DOCSLIB.ORG

Probabilistic Proof Systems Lecture Notes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Probabilistic Proof Systems Lecture Notes Probabilistic Pro of Systems { Lecture Notes Oded Goldreich Department of Computer Science and Applied Mathematics Weizmann Institute of Science, Rehovot, Israel. Septemb er 1996 Abstract Various typ es of probabilistic pro of systems have played a central role in the development of computer science in the last decade. In these notes, we concentrate on three such pro of systems | interactive proofs, zero-know ledge proofs, and probabilistic checkable proofs. Remark: These are lecture notes in the strict sense of the word. Surveys of mine on this sub ject can b e obtained from URL http://theory.lcs.mit.edu/ ~oded/pps.html. These notes were prepared for a series of lectures given in the Theory Student Seminar of the CS Department of UC-Berkeley. Visiting Miller Professor, Aug. { Sept. 1996, EECS Dept., UC{Berkeley. 0 1 Intro duction The glory given to the creativity required to nd pro ofs, makes us forget that it is the less glori- ed pro cedure of veri cation which gives pro ofs their value. Philosophically sp eaking, pro ofs are secondary to the veri cation pro cedure; whereas technically sp eaking, pro of systems are de ned in terms of their veri cation pro cedures. The notion of a veri cation pro cedure assumes the notion of computation and furthermore the notion of ecient computation. This implicit assumption is made explicit in the de nition of NP, in which ecient computation is asso ciated with deterministic p olynomial-time algorithms. Traditionally, NP is de ned as the class of NP-sets. Yet, each such NP-set can b e viewed as a pro of system. For example, consider the set of satis able Bo olean formulae. Clearly, a satisfying assignment for a formula constitutes an NP-pro of for the assertion \ is satis able" the veri cation pro cedure consists of substituting the variables of by the values assigned by and computing the value of the resulting Bo olean expression. The formulation of NP-pro ofs restricts the \e ective" length of pro ofs to b e p olynomial in length of the corresp onding assertions. However, longer pro ofs may b e allowed by padding the assertion with suciently many blank symb ols. So it seems that NP gives a satisfactory formulation of pro of systems with ecientveri cation pro cedures. This is indeed the case if one asso ciates ecient pro cedures with deterministic p olynomial-time algorithms. However, we can gain a lot if we are willing to take a somewhat non-traditional step and allow probabilistic veri cation pro cedures. In particular, Randomized and interactiveveri cation pro cedures, giving rise to interactive proof systems, seem much more p owerful i.e., \expressive" than their deterministic counterparts. Such randomized pro cedures allow the intro duction of zero-know ledge proofs which are of great theoretical and practical interest. NP-pro ofs can b e eciently transformed into a redundant form which o ers a trade-o between the numb er of lo cations examined in the NP-pro of and the con dence in its validity see probabilistical ly checkable proofs. In all ab ovementioned typ es of probabilistic pro of systems, explicit b ounds are imp osed on the computational complexity of the veri cation pro cedure, which in turn is p ersoni ed by the notion ofaveri er. Furthermore, in all these pro of systems, the veri er is allowed to toss coins and rule by statistical evidence. Thus, all these pro of systems carry a probability of error; yet, this probability is explicitly b ounded and, furthermore, can b e reduced by successive application of the pro of system. 2 Interactive Pro of Systems In light of the growing acceptability of randomized and distributed computations, it is only natural to asso ciate the notion of ecient computation with probabilistic and interactive p olynomial-time computations. This leads naturally to the notion of interactive pro of systems in which the veri ca- tion pro cedure is interactive and randomized, rather than b eing non-interactive and deterministic. Thus, a \pro of " in this context is not a xed and static ob ject but rather a randomized dynamic pro cess in which the veri er interacts with the prover. Intuitively, one may think of this interaction 1 as consisting of \tricky" questions asked by the veri er to which the prover has to reply \convinc- ingly". The ab ove discussion, as well as the actual de nition, makes explicit reference to a prover, whereas a prover is only implicit in the traditional de nitions of pro of systems e.g., NP-pro ofs. 2.1 The De nition Interaction: Going b eyond the uni-directional \interaction" of the NP-pro of system. If the veri er do es not toss coins then interaction can b e collapsed to a single message. computationall y unb ounded Prover: As in NP,we start by not considering the complexity of proving. probabilistic p olynomial-time Veri er: We maintain the paradigm that veri cation ought to b e easy, alas we allow random choices in our notion of easiness. Completeness and Soundness: We relax the traditional soundness condition by allowing small probability of b eing fo oled by false pro ofs. The probability is taken over the veri er's random choices. We still require \p erfect completeness"; that is, that correct statements are proven with probability 1. Error probability, b eing a parameter, can b e further reduced by successive rep eti- tions. Variations: Relaxing the \p erfect completeness" requirement yields a two-sided error variantof IP i.e., error probability allowed also in the completeness condition. Restricting the veri er to send only \random" i.e., uniformly chosen messages yields the restricted Arthur-Merlin interactive pro ofs aka public-coins interactive pro ofs. Alas, b oth variants are essentially as p owerful as the one ab ove. 2.2 An Example: interactive pro of of Graph Non-Isomorphism The problem: not known to b e in NP. Proving that two graphs are isomorphic can b e done by presenting an isomorphism, but howdoyou prove that no such isomorphism exists? The construction: the \two di erent ob ject proto col" { if you claim that two ob jects are di erent then you should b e able to tell which is which when I present them to you in random order. In the context of the Graph Non-Isomorphism interactive pro of, two supp osedly di erent ob jects are de ned by taking random isomorphic copies of each of the input graphs. If these graphs are indeed non-isomorphic then the ob jects are di erent the distributions have distinct supp ort else the ob jects are identical. 2.3 Interactive pro of of Non-Satis ability Arithmetization of Bo olean CNF formulae: Observe that the arithmetic expression is a low degree p olynomial. Observe that, in any case, the value of the arithmetic expression is b ounded. 2 Moving to a Finite Field: Whenever wecheck equalitybetween twointegers in [0;M], it suces to check equalitymodq, where q>M. The b ene t is that the arithmetic is now in a nite eld mo d q and so certain things are \nicer" e.g., uniformly selecting a value. Thus, proving that a CNF formula is not satis able reduces to proving equality of the following form X X x ; :::; x 0modq 1 n x =0;1 x =0;1 n 1 where isalow degree multi-variant p olynomial. The construction: stripping summations in iterations. In each iteration the prover is supp osed to supply the p olynomial describing the expression in one currently stripp ed variable. By the ab ove observation, this is a low degree p olynomial and so has a short description. The veri er checks that the p olynomial is of low degree, and that it corresp onds to the currentvalue b eing claimed i.e., p0 + p1 v . Next, the veri er randomly instantiates the variable, yielding a new value to b e claimed for the resulting expression i.e., v pr , for uniformly chosen r 2 GFq . The veri er sends the uniformly chosen instantiation to the prover. At the end of the last iteration, the veri er has a fully sp eci ed expression and can easily check it against the claimed value. Completeness of the ab ove: When the claim holds, the prover has no problem supplying the correct p olynomials, and this will lead the veri er to always accept. Soundness of the ab ove: It suces to b ound the probability that for a particular iteration the initial claim is false whereas the ending claim is correct. Both claims refer to the current summation expression b eing equal to the currentvalue, where `current' means either at the b eginning of the iteration or at its end. Let T b e the actual p olynomial representing the expression when stripping the currentvariable, and let pbeany p otential answer by the prover. Wemay assume that p0 + p1 v and that p is of low-degree as otherwise the veri er will reject. Using our hyp othesis that the initial claim is false, we know that T 0 + T 1 6 v . Thus, p and T are di erentlow-degree p olynomials and so they may agree on very few p oints. In case the veri er instantiation do es not happ en to b e one of these few p oints, the ending claim is false to o. Op en Problem 1 alternative pro of of coNP IP: Polynomials play a fundamental role in the above construction and this trend has even deepened in subsequent works on PCP.
Load more

Recommended publications
  • CS601 DTIME and DSPACE Lecture 5 Time and Space Functions: T, S
    CS601 DTIME and DSPACE Lecture 5 Time and Space functions: t, s : N → N+ Definition 5.1 A set A ⊆ U is in DTIME[t(n)] iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. A = L(M) ≡ w ∈ U M(w)=1 , and 2. ∀w ∈ U, M(w) halts within c · t(|w|) steps. Definition 5.2 A set A ⊆ U is in DSPACE[s(n)] iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. A = L(M), and 2. ∀w ∈ U, M(w) uses at most c · s(|w|) work-tape cells. (Input tape is “read-only” and not counted as space used.) Example: PALINDROMES ∈ DTIME[n], DSPACE[n]. In fact, PALINDROMES ∈ DSPACE[log n]. [Exercise] 1 CS601 F(DTIME) and F(DSPACE) Lecture 5 Definition 5.3 f : U → U is in F (DTIME[t(n)]) iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. f = M(·); 2. ∀w ∈ U, M(w) halts within c · t(|w|) steps; 3. |f(w)|≤|w|O(1), i.e., f is polynomially bounded. Definition 5.4 f : U → U is in F (DSPACE[s(n)]) iff there exists a deterministic, multi-tape TM, M, and a constant c, such that, 1. f = M(·); 2. ∀w ∈ U, M(w) uses at most c · s(|w|) work-tape cells; 3. |f(w)|≤|w|O(1), i.e., f is polynomially bounded. (Input tape is “read-only”; Output tape is “write-only”.
    [Show full text]
  • Interactive Proof Systems and Alternating Time-Space Complexity
    Theoretical Computer Science 113 (1993) 55-73 55 Elsevier Interactive proof systems and alternating time-space complexity Lance Fortnow” and Carsten Lund** Department of Computer Science, Unicersity of Chicago. 1100 E. 58th Street, Chicago, IL 40637, USA Abstract Fortnow, L. and C. Lund, Interactive proof systems and alternating time-space complexity, Theoretical Computer Science 113 (1993) 55-73. We show a rough equivalence between alternating time-space complexity and a public-coin interactive proof system with the verifier having a polynomial-related time-space complexity. Special cases include the following: . All of NC has interactive proofs, with a log-space polynomial-time public-coin verifier vastly improving the best previous lower bound of LOGCFL for this model (Fortnow and Sipser, 1988). All languages in P have interactive proofs with a polynomial-time public-coin verifier using o(log’ n) space. l All exponential-time languages have interactive proof systems with public-coin polynomial-space exponential-time verifiers. To achieve better bounds, we show how to reduce a k-tape alternating Turing machine to a l-tape alternating Turing machine with only a constant factor increase in time and space. 1. Introduction In 1981, Chandra et al. [4] introduced alternating Turing machines, an extension of nondeterministic computation where the Turing machine can make both existential and universal moves. In 1985, Goldwasser et al. [lo] and Babai [l] introduced interactive proof systems, an extension of nondeterministic computation consisting of two players, an infinitely powerful prover and a probabilistic polynomial-time verifier. The prover will try to convince the verifier of the validity of some statement.
    [Show full text]
  • Complexity Theory Lecture 9 Co-NP Co-NP-Complete
    Complexity Theory 1 Complexity Theory 2 co-NP Complexity Theory Lecture 9 As co-NP is the collection of complements of languages in NP, and P is closed under complementation, co-NP can also be characterised as the collection of languages of the form: ′ L = x y y <p( x ) R (x, y) { |∀ | | | | → } Anuj Dawar University of Cambridge Computer Laboratory NP – the collection of languages with succinct certificates of Easter Term 2010 membership. co-NP – the collection of languages with succinct certificates of http://www.cl.cam.ac.uk/teaching/0910/Complexity/ disqualification. Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 3 Complexity Theory 4 NP co-NP co-NP-complete P VAL – the collection of Boolean expressions that are valid is co-NP-complete. Any language L that is the complement of an NP-complete language is co-NP-complete. Any of the situations is consistent with our present state of ¯ knowledge: Any reduction of a language L1 to L2 is also a reduction of L1–the complement of L1–to L¯2–the complement of L2. P = NP = co-NP • There is an easy reduction from the complement of SAT to VAL, P = NP co-NP = NP = co-NP • ∩ namely the map that takes an expression to its negation. P = NP co-NP = NP = co-NP • ∩ VAL P P = NP = co-NP ∈ ⇒ P = NP co-NP = NP = co-NP • ∩ VAL NP NP = co-NP ∈ ⇒ Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 5 Complexity Theory 6 Prime Numbers Primality Consider the decision problem PRIME: Another way of putting this is that Composite is in NP.
    [Show full text]
  • On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs*
    On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs* Benny Applebaum† Eyal Golombek* Abstract We study the randomness complexity of interactive proofs and zero-knowledge proofs. In particular, we ask whether it is possible to reduce the randomness complexity, R, of the verifier to be comparable with the number of bits, CV , that the verifier sends during the interaction. We show that such randomness sparsification is possible in several settings. Specifically, unconditional sparsification can be obtained in the non-uniform setting (where the verifier is modelled as a circuit), and in the uniform setting where the parties have access to a (reusable) common-random-string (CRS). We further show that constant-round uniform protocols can be sparsified without a CRS under a plausible worst-case complexity-theoretic assumption that was used previously in the context of derandomization. All the above sparsification results preserve statistical-zero knowledge provided that this property holds against a cheating verifier. We further show that randomness sparsification can be applied to honest-verifier statistical zero-knowledge (HVSZK) proofs at the expense of increasing the communica- tion from the prover by R−F bits, or, in the case of honest-verifier perfect zero-knowledge (HVPZK) by slowing down the simulation by a factor of 2R−F . Here F is a new measure of accessible bit complexity of an HVZK proof system that ranges from 0 to R, where a maximal grade of R is achieved when zero- knowledge holds against a “semi-malicious” verifier that maliciously selects its random tape and then plays honestly.
    [Show full text]
  • The Complexity of Space Bounded Interactive Proof Systems
    The Complexity of Space Bounded Interactive Proof Systems ANNE CONDON Computer Science Department, University of Wisconsin-Madison 1 INTRODUCTION Some of the most exciting developments in complexity theory in recent years concern the complexity of interactive proof systems, defined by Goldwasser, Micali and Rackoff (1985) and independently by Babai (1985). In this paper, we survey results on the complexity of space bounded interactive proof systems and their applications. An early motivation for the study of interactive proof systems was to extend the notion of NP as the class of problems with efficient \proofs of membership". Informally, a prover can convince a verifier in polynomial time that a string is in an NP language, by presenting a witness of that fact to the verifier. Suppose that the power of the verifier is extended so that it can flip coins and can interact with the prover during the course of a proof. In this way, a verifier can gather statistical evidence that an input is in a language. As we will see, the interactive proof system model precisely captures this in- teraction between a prover P and a verifier V . In the model, the computation of V is probabilistic, but is typically restricted in time or space. A language is accepted by the interactive proof system if, for all inputs in the language, V accepts with high probability, based on the communication with the \honest" prover P . However, on inputs not in the language, V rejects with high prob- ability, even when communicating with a \dishonest" prover. In the general model, V can keep its coin flips secret from the prover.
    [Show full text]
  • EXPSPACE-Hardness of Behavioural Equivalences of Succinct One
    EXPSPACE-hardness of behavioural equivalences of succinct one-counter nets Petr Janˇcar1 Petr Osiˇcka1 Zdenˇek Sawa2 1Dept of Comp. Sci., Faculty of Science, Palack´yUniv. Olomouc, Czech Rep. [email protected], [email protected] 2Dept of Comp. Sci., FEI, Techn. Univ. Ostrava, Czech Rep. [email protected] Abstract We note that the remarkable EXPSPACE-hardness result in [G¨oller, Haase, Ouaknine, Worrell, ICALP 2010] ([GHOW10] for short) allows us to answer an open complexity ques- tion for simulation preorder of succinct one counter nets (i.e., one counter automata with no zero tests where counter increments and decrements are integers written in binary). This problem, as well as bisimulation equivalence, turn out to be EXPSPACE-complete. The technique of [GHOW10] was referred to by Hunter [RP 2015] for deriving EXPSPACE-hardness of reachability games on succinct one-counter nets. We first give a direct self-contained EXPSPACE-hardness proof for such reachability games (by adjust- ing a known PSPACE-hardness proof for emptiness of alternating finite automata with one-letter alphabet); then we reduce reachability games to (bi)simulation games by using a standard “defender-choice” technique. 1 Introduction arXiv:1801.01073v1 [cs.LO] 3 Jan 2018 We concentrate on our contribution, without giving a broader overview of the area here. A remarkable result by G¨oller, Haase, Ouaknine, Worrell [2] shows that model checking a fixed CTL formula on succinct one-counter automata (where counter increments and decre- ments are integers written in binary) is EXPSPACE-hard. Their proof is interesting and nontrivial, and uses two involved results from complexity theory.
    [Show full text]
  • Dspace 6.X Documentation
    DSpace 6.x Documentation DSpace 6.x Documentation Author: The DSpace Developer Team Date: 27 June 2018 URL: https://wiki.duraspace.org/display/DSDOC6x Page 1 of 924 DSpace 6.x Documentation Table of Contents 1 Introduction ___________________________________________________________________________ 7 1.1 Release Notes ____________________________________________________________________ 8 1.1.1 6.3 Release Notes ___________________________________________________________ 8 1.1.2 6.2 Release Notes __________________________________________________________ 11 1.1.3 6.1 Release Notes _________________________________________________________ 12 1.1.4 6.0 Release Notes __________________________________________________________ 14 1.2 Functional Overview _______________________________________________________________ 22 1.2.1 Online access to your digital assets ____________________________________________ 23 1.2.2 Metadata Management ______________________________________________________ 25 1.2.3 Licensing _________________________________________________________________ 27 1.2.4 Persistent URLs and Identifiers _______________________________________________ 28 1.2.5 Getting content into DSpace __________________________________________________ 30 1.2.6 Getting content out of DSpace ________________________________________________ 33 1.2.7 User Management __________________________________________________________ 35 1.2.8 Access Control ____________________________________________________________ 36 1.2.9 Usage Metrics _____________________________________________________________
    [Show full text]
  • Simple Doubly-Efficient Interactive Proof Systems for Locally
    Electronic Colloquium on Computational Complexity, Revision 3 of Report No. 18 (2017) Simple doubly-efficient interactive proof systems for locally-characterizable sets Oded Goldreich∗ Guy N. Rothblumy September 8, 2017 Abstract A proof system is called doubly-efficient if the prescribed prover strategy can be implemented in polynomial-time and the verifier’s strategy can be implemented in almost-linear-time. We present direct constructions of doubly-efficient interactive proof systems for problems in P that are believed to have relatively high complexity. Specifically, such constructions are presented for t-CLIQUE and t-SUM. In addition, we present a generic construction of such proof systems for a natural class that contains both problems and is in NC (and also in SC). The proof systems presented by us are significantly simpler than the proof systems presented by Goldwasser, Kalai and Rothblum (JACM, 2015), let alone those presented by Reingold, Roth- blum, and Rothblum (STOC, 2016), and can be implemented using a smaller number of rounds. Contents 1 Introduction 1 1.1 The current work . 1 1.2 Relation to prior work . 3 1.3 Organization and conventions . 4 2 Preliminaries: The sum-check protocol 5 3 The case of t-CLIQUE 5 4 The general result 7 4.1 A natural class: locally-characterizable sets . 7 4.2 Proof of Theorem 1 . 8 4.3 Generalization: round versus computation trade-off . 9 4.4 Extension to a wider class . 10 5 The case of t-SUM 13 References 15 Appendix: An MA proof system for locally-chracterizable sets 18 ∗Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel.
    [Show full text]
  • On the Existence of Extractable One-Way Functions
    On the Existence of Extractable One-Way Functions The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Bitansky, Nir et al. “On the Existence of Extractable One-Way Functions.” SIAM Journal on Computing 45.5 (2016): 1910–1952. © 2016 by SIAM As Published http://dx.doi.org/10.1137/140975048 Publisher Society for Industrial and Applied Mathematics Version Final published version Citable link http://hdl.handle.net/1721.1/107895 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. SIAM J. COMPUT. c 2016 Society for Industrial and Applied Mathematics Vol. 45, No. 5, pp. 1910{1952 ∗ ON THE EXISTENCE OF EXTRACTABLE ONE-WAY FUNCTIONS NIR BITANSKYy , RAN CANETTIz , OMER PANETHz , AND ALON ROSENx Abstract. A function f is extractable if it is possible to algorithmically \extract," from any adversarial program that outputs a value y in the image of f , a preimage of y. When combined with hardness properties such as one-wayness or collision-resistance, extractability has proven to be a powerful tool. However, so far, extractability has not been explicitly shown. Instead, it has only been considered as a nonstandard knowledge assumption on certain functions. We make headway in the study of the existence of extractable one-way functions (EOWFs) along two directions. On the negative side, we show that if there exist indistinguishability obfuscators for circuits, then there do not exist EOWFs where extraction works for any adversarial program with auxiliary input of unbounded polynomial length.
    [Show full text]
  • Glossary of Complexity Classes
    App endix A Glossary of Complexity Classes Summary This glossary includes selfcontained denitions of most complexity classes mentioned in the b o ok Needless to say the glossary oers a very minimal discussion of these classes and the reader is re ferred to the main text for further discussion The items are organized by topics rather than by alphab etic order Sp ecically the glossary is partitioned into two parts dealing separately with complexity classes that are dened in terms of algorithms and their resources ie time and space complexity of Turing machines and complexity classes de ned in terms of nonuniform circuits and referring to their size and depth The algorithmic classes include timecomplexity based classes such as P NP coNP BPP RP coRP PH E EXP and NEXP and the space complexity classes L NL RL and P S P AC E The non k uniform classes include the circuit classes P p oly as well as NC and k AC Denitions and basic results regarding many other complexity classes are available at the constantly evolving Complexity Zoo A Preliminaries Complexity classes are sets of computational problems where each class contains problems that can b e solved with sp ecic computational resources To dene a complexity class one sp ecies a mo del of computation a complexity measure like time or space which is always measured as a function of the input length and a b ound on the complexity of problems in the class We follow the tradition of fo cusing on decision problems but refer to these problems using the terminology of promise problems
    [Show full text]
  • The Weakness of CTC Qubits and the Power of Approximate Counting
    The weakness of CTC qubits and the power of approximate counting Ryan O'Donnell∗ A. C. Cem Sayy April 7, 2015 Abstract We present results in structural complexity theory concerned with the following interre- lated topics: computation with postselection/restarting, closed timelike curves (CTCs), and approximate counting. The first result is a new characterization of the lesser known complexity class BPPpath in terms of more familiar concepts. Precisely, BPPpath is the class of problems that can be efficiently solved with a nonadaptive oracle for the Approximate Counting problem. Similarly, PP equals the class of problems that can be solved efficiently with nonadaptive queries for the related Approximate Difference problem. Another result is concerned with the compu- tational power conferred by CTCs; or equivalently, the computational complexity of finding stationary distributions for quantum channels. Using the above-mentioned characterization of PP, we show that any poly(n)-time quantum computation using a CTC of O(log n) qubits may as well just use a CTC of 1 classical bit. This result essentially amounts to showing that one can find a stationary distribution for a poly(n)-dimensional quantum channel in PP. ∗Department of Computer Science, Carnegie Mellon University. Work performed while the author was at the Bo˘gazi¸ciUniversity Computer Engineering Department, supported by Marie Curie International Incoming Fellowship project number 626373. yBo˘gazi¸ciUniversity Computer Engineering Department. 1 Introduction It is well known that studying \non-realistic" augmentations of computational models can shed a great deal of light on the power of more standard models. The study of nondeterminism and the study of relativization (i.e., oracle computation) are famous examples of this phenomenon.
    [Show full text]
  • Independence of P Vs. NP in Regards to Oracle Relativizations. · 3 Then
    Independence of P vs. NP in regards to oracle relativizations. JERRALD MEEK This is the third article in a series of four articles dealing with the P vs. NP question. The purpose of this work is to demonstrate that the methods used in the first two articles of this series are not affected by oracle relativizations. Furthermore, the solution to the P vs. NP problem is actually independent of oracle relativizations. Categories and Subject Descriptors: F.2.0 [Analysis of Algorithms and Problem Complex- ity]: General General Terms: Algorithms, Theory Additional Key Words and Phrases: P vs NP, Oracle Machine 1. INTRODUCTION. Previously in “P is a proper subset of NP” [Meek Article 1 2008] and “Analysis of the Deterministic Polynomial Time Solvability of the 0-1-Knapsack Problem” [Meek Article 2 2008] the present author has demonstrated that some NP-complete problems are likely to have no deterministic polynomial time solution. In this article the concepts of these previous works will be applied to the relativizations of the P vs. NP problem. It has previously been shown by Hartmanis and Hopcroft [1976], that the P vs. NP problem is independent of Zermelo-Fraenkel set theory. However, this same discovery had been shown by Baker [1979], who also demonstrates that if P = NP then the solution is incompatible with ZFC. The purpose of this article will be to demonstrate that the oracle relativizations of the P vs. NP problem do not preclude any solution to the original problem. This will be accomplished by constructing examples of five out of the six relativizing or- acles from Baker, Gill, and Solovay [1975], it will be demonstrated that inconsistent relativizations do not preclude a proof of P 6= NP or P = NP.
    [Show full text]